If $e^y+xy=e^x$, then $y_2\left(0\right)=$

$e^y+xy=e^x\cdots \cdots \left(1\right)$

$x=0\Rightarrow e^y=1\Rightarrow y=0$     

differentiate with respect to $x$

$e^y.y^1+(y+x{y}^1)=e^x\cdots \cdots \left(2\right)$

$x=0,y=0$

$e^0.y^1\left(0\right)+(0+0)=e^0=1$

$y^1\left(0\right)=1$

$e^y.y^{11}+e^y.y^1.y^1+y^1+x{y}^{11}+y^1=e^x$  (differentiate with respect to $x$ in (2))

$\text{put\hspace{0.17em}\hspace{0.17em}}x=0,y=0,y^1\left(0\right)=1$

$1.y^{11}\left(0\right)+e^0\mathrm{.1.1}+1+0+1=e^0$

$y^{11}\left(0\right)+3=1$

$y^{11}\left(0\right)=-2$